Research On Well-posedness For Two Classes Of Nonlinear Wave Equations With Variable Exponent Source

The thesis considers two classes of wave equations with variable exponential source based on the potential well theorey.One is a wave equation with nonlinear weak damping term and variable exponential source term,the other is a wave equation with a strong weak damping term,m-Laplacian term,and variable exponential source term.The above equations can be used to describe a variety of physical phenomena,such as: electrorheological fluids,filtration processes in porous media and so on.The thesis systematically considers the global existence,asymptotic behavior and finite-time blow up of solutions for the two classes of equations at the whole initial energy.The study aims to reveal the influence of different indicators,different structures,and different factors on the well-posed properties of solutions.The first part of this thesis studies the initial boundary value problem of a class of wave equation with nonlinear weak damping term and variable exponential source term.Firstly,the thesis gives the total energy function,potential energy function,Nehari function,potential well depth and some important lemmas corresponding to the problem.Secondly,the stable set and unstable set at subcritical and critical initial energy are given.are given respectively,then the global existence,the asymptotic behavior and the global non-existence of solutions at the subcritical initial energy and the critical initial energy are obtained.In addition,the thesis considers the global non-existence of solutions at the supcritical initial energy.Finally,the lower bound of the blowup time is also estimated.The second part of this thesis considers the initial boundary value problem of a class of wave equation with strong and weak damping terms,m-Laplacian term and variable exponential source term.Firstly,the thesis gives the existence and uniqueness of the local solution.Then using the variational relationship between the potential energy functional and Nehari functional to give the potential well depth,which divides the space of the initial data.Secondly,the global existence and non-existence of solutions at the subcritical initial energy and the critical initial energy are considered by using the method of Galerkin and concave function.At the same time,this thesis discusses the long-term behavior of solutions,and get the conclusion that the energy decays exponentially.In addition,the global non-existence of solutions at the supercritical initial energy is given by using the improved concave function method.At last,the thesis estimates the lower bound of the blowup time by controlling the differential inequality.